Finite Group Databasebeta

← C3×#71755 #11775259

Subgroups — C3×#71755 #11775259

Order 1152.

601 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#529C211yesmaximal, characteristic
2C3×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C3×#4926C211yesmaximal, characteristic, complemented
4C3×#4538C212yesmaximal, complemented
5C3×#4537C212yesmaximal
6C3×#1509—31nomaximal
7#71755C311yesmaximal, characteristic, direct product factor
8C32×#126—21no—
9C32×#125C411yescharacteristic, complemented
10C32×#124C411yescharacteristic, complemented
11C32×#123—21no—
12C32×C22⋊Q8C2211yescharacteristic
13C32×#121C2211yescharacteristic
14C32×C4×Q8C411yescharacteristic, complemented
15C32×C4×D4—21no—
16C32×C4×D4C411yescharacteristic, complemented
17C32×#117—21no—
18C2×C32×C42C2211yescharacteristic
19C2×C3×C4⋊Dic3—21no—
20C2×C3×C4×Dic3—21no—
21C3×C3⋊#95C2212yes—
22C3×#830C2212yes—
23C3×C4⋊(C3⋊C8)—21no—
24C3×#803—21no—
25C3×#529—21no—
26C3×#529D311yescharacteristic
27C3×#529C2×C311yescharacteristic
28C3×#432—31no—
29C3×C4×C4⋊C4—31no—
30C4×C4⋊Dic3C2×C311yescharacteristic, complemented
31#4926C2×C311yescharacteristic, complemented
32C3×#328—32no—
33C3×#327—32no—
34#4538C2×C312yescomplemented
35#4537C2×C312yes—
36C2×C32×Q8C2×C411yescharacteristic
37C2×C32×D4—21no—
38C2×C32×D4C2×C411yescharacteristic
39C22×C32×C4—21no—
40C22×C32×C4D411yescharacteristic
41C22×C32×C4D411yescharacteristic
42C22×C32×C4C2311yescharacteristic
43C22×C3×Dic3—22no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4—21no—
48C32×C4⋊C4C2×C411yescharacteristic
49C32×C4⋊C4C2×C411yescharacteristic
50C32×C22⋊C4—21no—
51C32×C22⋊C4—21no—
52C32×C22⋊C4—21no—
53C32×C22⋊C4—21no—
54C32×C22⋊C4—21no—
55C32×C42—21no—
56C32×C42C2×C411yescharacteristic
57C32×C42C2×C411yescharacteristic
58C32×C42D411yescharacteristic
59C32×C42D411yescharacteristic
60C3×C4⋊Dic3—24no—
61C3×C4×Dic3—42no—
62C2×C3×C3⋊C8—22no—
63#1509—31noSylow
64C3×#126—22no—
65C3×#126—21no—
66C3×#126—21no—
67C3×#125—21no—
68C3×#125Dic311yescharacteristic, complemented
69C3×#125C3×C411yescharacteristic, complemented
70C3×#124—21no—
71C3×#124Dic311yescharacteristic, complemented
72C3×#124C3×C411yescharacteristic, complemented
73C3×#123—22no—
74C3×#123—21no—
75C3×#123—21no—
76C3×C22⋊Q8—21no—
77C3×C22⋊Q8C2×D311yescharacteristic
78C3×C22⋊Q8C22×C311yescharacteristic
79C3×#121—21no—
80C3×#121C2×D311yescharacteristic
81C3×#121C22×C311yescharacteristic
82C3×C4×Q8—21no—
83C3×C4×Q8Dic311yescharacteristic, complemented
84C3×C4×Q8C3×C411yescharacteristic, complemented
85C3×C4×D4—22no—
86C3×C4×D4—21no—
87C3×C4×D4—21no—
88C3×C4×D4—21no—
89C3×C4×D4Dic311yescharacteristic, complemented
90C3×C4×D4C3×C411yescharacteristic, complemented
91C3×#117—22no—
92C3×#117—21no—
93C3×#117—21no—
94C2×C3×C4⋊C4—61no—
95C2×C3×C42—21no—
96C2×C3×C42—61no—
97C2×C3×C42C2×D311yescharacteristic
98C2×C3×C42C22×C311yescharacteristic
99C2×C4⋊Dic3—21no—
100C2×C4×Dic3—21no—
101C3×C4⋊C8—61no—
102C3×C22⋊C8—32no—
103C3×#97—61no—
104C3×#95—32no—
105C3⋊#95C22×C312yes—
106#830C22×C312yes—
107C4⋊(C3⋊C8)—21no—
108#803—21no—
109C23×C32—21no—
110C23×C32C2×D411yescharacteristic
111C32×Q8—41no—
112C32×D4—41no—
113C32×D4—41no—
114C32×D4—41no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—22no—
121C2×C32×C4—41no—
122C2×C32×C4—42no—
123C2×C32×C4C22⋊C411yescharacteristic
124C2×C32×C4C22⋊C411yescharacteristic
125C2×C32×C4C22⋊C411yescharacteristic
126C2×C32×C4C22⋊C411yescharacteristic
127C2×C32×C4C22×C411yescharacteristic
128C2×C32×C4C2×D411yescharacteristic
129C2×C3×Dic3—24no—
130C2×C3×Dic3—42no—
131C2×C3×Dic3—42no—
132C3×C3⋊C8—42no—
133#529C3×D311yescharacteristic
134#432—31no—
135C4×C4⋊C4—31no—
136#328—32no—
137#327—32no—
138C2×C3×Q8—21no—
139C2×C3×Q8C2×Dic311yescharacteristic
140C2×C3×Q8C2×C3×C411yescharacteristic
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4C2×Dic311yescharacteristic
146C2×C3×D4C2×C3×C411yescharacteristic
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—62no—
154C22×C3×C4#6711yescharacteristic
155C22×C3×C4#6711yescharacteristic
156C22×C3×C4C3×D411yescharacteristic
157C22×C3×C4C3×D411yescharacteristic
158C22×C3×C4C22×D311yescharacteristic
159C22×C3×C4C23×C311yescharacteristic
160C22×Dic3—22no—
161C2×C3×C8—62no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—64no—
177C3×C4⋊C4C2×Dic311yescharacteristic
178C3×C4⋊C4C2×Dic311yescharacteristic
179C3×C4⋊C4C2×C3×C411yescharacteristic
180C3×C4⋊C4C2×C3×C411yescharacteristic
181C3×C22⋊C4—22no—
182C3×C22⋊C4—22no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—22no—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—21no—
187C3×C22⋊C4—21no—
188C3×C22⋊C4—21no—
189C3×C22⋊C4—21no—
190C3×C22⋊C4—21no—
191C3×C22⋊C4—21no—
192C3×C22⋊C4—21no—
193C3×C22⋊C4—21no—
194C3×C22⋊C4—21no—
195C3×C22⋊C4—21no—
196C3×C42—22no—
197C3×C42—21no—
198C3×C42—21no—
199C3×C42—21no—
200C3×C42—21no—
201C3×C42—21no—
202C3×C42—21no—
203C3×C42—122no—
204C3×C42C2×Dic311yescharacteristic
205C3×C42C2×Dic311yescharacteristic
206C3×C42#6711yescharacteristic
207C3×C42#6711yescharacteristic
208C3×C42C2×C3×C411yescharacteristic
209C3×C42C2×C3×C411yescharacteristic
210C3×C42C3×D411yescharacteristic
211C3×C42C3×D411yescharacteristic
212C4⋊Dic3—24no—
213C4×Dic3—42no—
214C2×C3⋊C8—22no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32C2×C22⋊C411yescharacteristic
221C22×C32AGL(1,ℤ/8)11yescharacteristic
222C22×C32#13711yescharacteristic
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—21no—
228C32×C4—41no—
229C32×C4—41no—
230C32×C4—41no—
231C32×C4#10412yes—
232C3×Dic3—44no—
233#126—21no—
234#125C3×Dic311yescharacteristic, complemented
235#124C3×Dic311yescharacteristic, complemented
236#123—21no—
237C22⋊Q8C2×C3×D311yescharacteristic
238#121C2×C3×D311yescharacteristic
239C4×Q8C3×Dic311yescharacteristic, complemented
240C4×D4—21no—
241C4×D4C3×Dic311yescharacteristic, complemented
242#117—21no—
243C2×C4⋊C4—61no—
244C2×C42—61no—
245C2×C42C2×C3×D311yescharacteristic
246C4⋊C8—61no—
247C22⋊C8—32no—
248#97—61no—
249#95—32no—
250C23×C3—22no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3C2×#6711yescharacteristic
255C23×C3C2×C3×D411yescharacteristic
256C3×Q8—41no—
257C3×Q8—41no—
258C3×Q8—42no—
259C3×D4—41no—
260C3×D4—41no—
261C3×D4—42no—
262C3×D4—42no—
263C3×D4—42no—
264C3×D4—41no—
265C3×D4—41no—
266C3×D4—41no—
267C3×D4—41no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—44no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—64no—
299C2×C3×C4—122no—
300C2×C3×C4—122no—
301C2×C3×C4#21711yescharacteristic
302C2×C3×C4#21711yescharacteristic
303C2×C3×C4#21711yescharacteristic
304C2×C3×C4#21711yescharacteristic
305C2×C3×C4C3×C22⋊C411yescharacteristic
306C2×C3×C4C3×C22⋊C411yescharacteristic
307C2×C3×C4C3×C22⋊C411yescharacteristic
308C2×C3×C4C3×C22⋊C411yescharacteristic
309C2×C3×C4C22×Dic311yescharacteristic
310C2×C3×C4C2×#6711yescharacteristic
311C2×C3×C4C22×C3×C411yescharacteristic
312C2×C3×C4C2×C3×D411yescharacteristic
313C2×Dic3—24no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C3×C8—122no—
317C3⋊C8—42no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—81no—
321C2×C32#41011yescharacteristic
322C2×C32#41711yescharacteristic
323C2×C32C2×#10411yescharacteristic
324C2×Q8C2×C3×Dic311yescharacteristic
325C2×D4—21no—
326C2×D4C2×C3×Dic311yescharacteristic
327C22×C4—21no—
328C22×C4—62no—
329C22×C4C3×#6711yescharacteristic
330C22×C4C3×#6711yescharacteristic
331C22×C4C22×C3×D311yescharacteristic
332C2×C8—62no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—64no—
338C4⋊C4C2×C3×Dic311yescharacteristic
339C4⋊C4C2×C3×Dic311yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C42—21no—
346C42—122no—
347C42C2×C3×Dic311yescharacteristic
348C42C2×C3×Dic311yescharacteristic
349C42C3×#6711yescharacteristic
350C42C3×#6711yescharacteristic
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3#94911yescharacteristic
370C22×C3#95111yescharacteristic
371C22×C3C2×#21711yescharacteristic
372C22×C3C2×C3×C22⋊C411yescharacteristic
373C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
374C22×C3C3×#13711yescharacteristic
375C3×C4—22no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—22no—
387C3×C4—41no—
388C3×C4—41no—
389C3×C4—41no—
390C3×C4—41no—
391C3×C4—42no—
392C3×C4—42no—
393C3×C4—42no—
394C3×C4—41no—
395C3×C4—41no—
396C3×C4—41no—
397C3×C4—41no—
398C3×C4—41no—
399C3×C4—41no—
400C3×C4—124no—
401C3×C4#83712yes—
402C3×C4C3×#10412yes—
403Dic3—44no—
404C32#150911yescharacteristic, complemented, Sylow
405C23—21no—
406C23C2×C3×#6711yescharacteristic
407Q8—41no—
408D4—41no—
409D4—41no—
410D4—41no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—64no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4C3×#21711yescharacteristic
423C2×C4C3×#21711yescharacteristic
424C2×C4C3×#21711yescharacteristic
425C2×C4C3×#21711yescharacteristic
426C2×C4C22×C3×Dic311yescharacteristic
427C2×C4C2×C3×#6711yescharacteristic
428C8—122no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—41no—
437C2×C3—41no—
438C2×C3—82no—
439C2×C3—81no—
440C2×C3—81no—
441C2×C3#523211yescharacteristic
442C2×C3C2×#83711yescharacteristic
443C2×C3#524011yescharacteristic
444C2×C3C3×#41011yescharacteristic
445C2×C3C3×#41711yescharacteristic
446C2×C3C2×C3×#10411yescharacteristic
447C22—21no—
448C22—21no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22C3×#94911yescharacteristic
453C22C3×#95111yescharacteristic
454C22C2×C3×#21711yescharacteristic
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4—124no—
464C4C3×#83712yes—
465C3—21no—
466C3#7175511yescharacteristic, direct product factor
467C3C3×#150911yescharacteristic, complemented
468C2—21no—
469C2—21no—
470C2—81no—
471C2C3×#523211yescharacteristic
472C2C2×C3×#83711yescharacteristic
473C2C3×#524011yescharacteristic
subgroupG/H
centerC22×C3C2×#217
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#529C2
socleC22×C32C2×C22⋊C4
prime radicalC32×#121C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#529D3
hypocenterC3C3×#1509
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C3×#529

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×C4×C4⋊Dic3C2semidirect
3. C3×#4926C2semidirect
4. C3×#4538C2semidirect
7. #71755C3direct (N × H)
9. C32×#125C4semidirect
10. C32×#124C4semidirect
14. C32×C4×Q8C4semidirect
16. C32×C4×D4C4semidirect
30. C4×C4⋊Dic3C2×C3semidirect
31. #4926C2×C3semidirect
34. #4538C2×C3semidirect
68. C3×#125Dic3semidirect
69. C3×#125C3×C4semidirect
71. C3×#124Dic3semidirect
72. C3×#124C3×C4semidirect
83. C3×C4×Q8Dic3semidirect
84. C3×C4×Q8C3×C4semidirect
89. C3×C4×D4Dic3semidirect
90. C3×C4×D4C3×C4semidirect
234. #125C3×Dic3semidirect
235. #124C3×Dic3semidirect
239. C4×Q8C3×Dic3semidirect
241. C4×D4C3×Dic3semidirect
404. C32#1509semidirect
466. C3#71755direct (N × H)
467. C3C3×#1509semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C32×#5292
C3×C4×C4⋊Dic32
C3×#49262
C3×#45382
C3×#45372
C3×#15093
#717553